Theorem grpsubfvalg 13627

Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.)

Hypotheses

Ref	Expression
grpsubval.b	|- B = ( Base ` G )
grpsubval.p	|- .+ = ( +g ` G )
grpsubval.i	|- I = ( invg ` G )
grpsubval.m	|- .- = ( -g ` G )

Assertion

Ref	Expression
grpsubfvalg	|- ( G e. V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )

Distinct variable groups:   x, y, B   x, G, y   x, I, y   x, .+ , y

Allowed substitution hints:   .- ( x, y)   V( x, y)

Proof of Theorem grpsubfvalg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpsubval.m	. 2 |- .- = ( -g ` G )
2		df-sbg 13587	. . 3 |- -g = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) )
3		fveq2 5639	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpsubval.b	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2282	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5639	. . . . . 6 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		grpsubval.p	. . . . . 6 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2282	. . . . 5 |- ( g = G -> ( +g ` g ) = .+ )
9		eqidd 2232	. . . . 5 |- ( g = G -> x = x )
10		fveq2 5639	. . . . . . 7 |- ( g = G -> ( invg ` g ) = ( invg ` G ) )
11		grpsubval.i	. . . . . . 7 |- I = ( invg ` G )
12	10, 11	eqtr4di 2282	. . . . . 6 |- ( g = G -> ( invg ` g ) = I )
13	12	fveq1d 5641	. . . . 5 |- ( g = G -> ( ( invg ` g ) ` y ) = ( I ` y ) )
14	8, 9, 13	oveq123d 6038	. . . 4 |- ( g = G -> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) = ( x .+ ( I ` y ) ) )
15	5, 5, 14	mpoeq123dv 6082	. . 3 |- ( g = G -> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
16		elex 2814	. . 3 |- ( G e. V -> G e. _V )
17		basfn 13140	. . . . . 6 |- Base Fn _V
18		funfvex 5656	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
19	18	funfni 5432	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
20	17, 16, 19	sylancr 414	. . . . 5 |- ( G e. V -> ( Base ` G ) e. _V )
21	4, 20	eqeltrid 2318	. . . 4 |- ( G e. V -> B e. _V )
22		mpoexga 6376	. . . 4 |- ( ( B e. _V /\ B e. _V ) -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V )
23	21, 21, 22	syl2anc 411	. . 3 |- ( G e. V -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V )
24	2, 15, 16, 23	fvmptd3 5740	. 2 |- ( G e. V -> ( -g ` G ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
25	1, 24	eqtrid 2276	1 |- ( G e. V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   Basecbs 13081   +g cplusg 13159   invgcminusg 13583   -gcsg 13584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-sbg 13587

This theorem is referenced by:  grpsubval  13628  grpsubf  13661  grpsubpropdg  13686  grpsubpropd2  13687